Graph theory

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Graph theory

• A graph consists of
• set of vertices
• A set of edges connecting vertex pair
• Incidence matrix which edges are connected

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The incidence matrix of a graph gives the
(0,1)-matrix which has a row for each vertex and
column for each edge, and (v,e)1 iff vertex v is
incident upon edge e
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These are all equivalent
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Euler and the Konigsberg bridges
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Types of graphs

• Eulerian circuit that traverses each edge
  exactly once
• Which graphs possess Euler circuits?

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Problem does this graph have an Euler cycle?
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Theorem If every vertex has even degree then
there is an Eulerian path
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What is a theorem?

• A statement that no one can understand
• A statement that only a mathematician can
  understand
• A statement that can be verified from first
  principles
• A statement that is always true

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Heuristic argument

• An argument that appeals to intuition, but may
  not be compelling by itself.
• In the case of the Eulerian graph theorem, think
  of the vertex as a room and the edges as hallways
  connecting rooms.
• If you leave using one hallway then you have to
  return using a different one.
• Induction argument

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Hamiltonian graph
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Hamiltons puzzle find a path in the
dodecahedron graph that traverses each vertex
exactly once
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Is the following graph Hamiltonian?
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Is the following graph Hamiltonian?
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Petersen graph symmetry
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Graph colorings
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Other types of graphs
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Other properties

• Diameter
• Girth
• Chromatic number
• etc

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Which continent is this?
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Bosss dilemna

• Six employees, A,B,C,D,E,F
• Some do not get along with others
• Find smallest number of compatible work groups

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What does this graph have to do with the Bosss
dilemma?
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Complementary graph
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Complete subgraph

• Subgraph vertices subset of vertex set, edges
  subset of edge set
• Complete every vertex is connected to every
  other vertex.

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Complementary graph
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Handshakes, part 2

• There are several men and 15 women in a room.
  Each man shakes hands with exactly 6 women, and
  each woman shakes hands with exactly 8 men.
• How many men are in the room?

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Visualize whirled peas

• Samantha the sculptress wishes to make world
  peace sculpture based on the following idea she
  will sculpt 7 pillars, one for each continent,
  placing them in circle. Then she will string
  gold thread between the pillars so that each
  pillar is connected to exactly 3 others.
• Can Samantha do this?

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Some additional exercises in graph theory

• There are 7 guests at a formal dinner party. The
  host wishes each person to shake hands with each
  other person, for a total of 21 handshakes,
  according to
• Each handshake should involve someone from the
  previous handshake
• No person should be involved in 3 consecutive
  handshakes
• Is this possible?

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Camelot

• King Arthur and his knights wish to sit at the
  round table every evening in such a way that each
  person has different neighbors on each occasion.
  If KA has 10 knights, for how long can he do
  this?
• Suppose he wants to do this for 7 nights. How
  many knights does he need, at a minimum?

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The PNP problem

• The question is whether, for all problems for
  which a computer can verify a given solution
  quickly (that is, in polynomial time), it can
  also find that solution quickly. This is
  generally considered the most important open
  question in theoretical computer science as it
  has far-reaching consequences in mathematics,
  philosophy and cryptography.

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P vs NP classes
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• P NP? asks if 'yes'-answers to a
  'yes'-or-'no'-question can be verified "quickly"
  (in polynomial time), can the answers themselves
  also be computed quickly?
• Example subset-sum problem
• "easy" to verify,
• believed (but not proven) "difficult" to compute.
• Given a set of integers, does some subset of
  them sum to 0?
• Example -2, -3, 15, 14, 7, -10
• -2, -3, -10, 15 adds up to zero
• finding such a subset in the first place could
  take much longer.
• this problem is in NP.

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• IF P NP then problems like the subset-sum
  problem are as "easy" to compute as to verify.
• If P does not equal NP, some NP problems are
  substantially "harder" to compute than to verify.

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• This is item 1
• This is item 2
• This is item 3

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Example Primality

• Testing whether a number is prime can be done in
  polynomial time (Shor, 1994) on a quantum
  computer (qubits), but not on a classical
  computer (bits)
• On a classical computer, the problem is
  subexponential

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NP-complete

• NP-hard problems are those to which any problem
  in NP can be reduced in polynomial time.
• For instance, the decision problem version of
  the traveling salesman problem is NP-complete, so
  any instance of any problem in NP can be
  transformed mechanically into an instance of the
  traveling salesman problem, in polynomial time.
• The traveling salesman problem is one of many
  such NP-complete problems. If any NP-complete
  problem is in P, then it would follow that P
  NP. Unfortunately, many important problems have
  been shown to be NP-complete and as of 2008, not
  a single fast algorithm for any of them is known.

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• It is not obvious that NP-complete problems
  exist.
• Given a description of a Turing machine M
  guaranteed to halt in polynomial time, does there
  exist a polynomial-size input that M will accept?
• This is in NP given an input, it is simple to
  check whether or not M accepts the input by
  simulating M
• It is NP-hard the verifier for any particular
  instance of a problem in NP can be encoded as a
  polynomial-time machine M that takes the solution
  to be verified as input.
• The question of whether the instance is a yes or
  no instance is determined by whether a valid
  input exists.

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Is P equal to NP?

• In a 2002 poll of 100 researchers, 61 believed
  the answer is no, 9 believed the answer is yes,
  22 were unsure, and 8 believed the question may
  impossible to prove or disprove.

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• Most computer scientists believe that P?NP.
• Key reason no one has been able to find a
  polynomial-time algorithm for any of the more
  than 3000 NP-complete problems
• These algorithms were sought long before the
  concept of NP-completeness was even known
• The result P NP would imply many other
  startling results that are currently believed to
  be false, such as NP co-NP and P PH.

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Millennium Prize Problems

• The Millennium Prize Problems are seven problems
  in mathematics that were stated by the Clay
  Mathematics Institute in 2000. Currently, six of
  the problems remain unsolved. A correct solution
  to each problem results in a US1,000,000 prize
  (sometimes called a Millennium Prize) being
  awarded by the institute. Only the Poincaré
  conjecture has been solved, but the solver
  Grigori Perelman has not pursued the conditions
  necessary to claim the prize.

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Turing machine

• A Turing machine is a basic, abstract
  symbol-manipulating device that can be adapted to
  simulate the logic of any computer algorithm.
• Described in 1936 by Alan Turing.
• Not intended as a practical computing technology,
  rather as a thought experiment about the limits
  of mechanical computation.
• Never actually constructed but their abstract
  properties yields many insights into computer
  science and complexity theory.

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Turing machine (details)

• TAPE divided into cells, one next to the other.
  Each cell contains a symbol from some finite
  alphabet. The alphabet contains a special blank
  symbol (here written as '0') and one or more
  other symbols. The tape is assumed to be
  arbitrarily extendable to the left and to the
  right, i.e., the Turing machine is always
  supplied with as much tape as it needs for its
  computation. Cells that have not been written to
  before are assumed to be filled with the blank
  symbol
• A HEAD that can read and write symbols on the
  tape and move the tape left and right one (and
  only one) cell at a time. In some models the head
  moves and the tape is stationary.
• A finite ACTION TABLE given the current
  state(qi) and the symbol(aj) it is reading, tells
  the machine to do the following in sequence (for
  the 5-tuple models) either erase or write a
  symbol (instead of aj written aj1), and then
  move the head 'L' for one step left or 'R' for
  one step right or 'H' for staying put assume
  the same or a new state as prescribed (go to
  state qi1).A state register that stores the
  state of the Turing table, one of finitely many.

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Universal Turing Machine (UTM)

• A UTM is able to simulate any other Turing.
• Church-Turing thesis Turing machines indeed
  capture the informal notion of effective method
  in logic and mathematics, and provide a precise
  definition of an algorithm or 'mechanical
  procedure'.

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Schematics
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The Busy Beaver
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